### A flat plane that is not the limit of periodic flat planes.

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We prove that an Artin-Tits group of type $\tilde{C}$ is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the “generated group” method. This answers, in this particular case, a general question on Artin-Tits groups, gives a new presentation of an Artin-Tits group of type $\tilde{C}$, and has consequences for the word problem, the computation of some centralizers or the triviality of the center. A key point of the proof...

In this expository article we use topological ideas, notably compactness, to establish certain basic properties of orderable groups. Many of the properties we shall discuss are well-known, but I believe some of the proofs are new. These will be used, in turn, to prove some orderability results, including the left-orderability of the group of PL homeomorphisms of a surface with boundary, which are fixed on at least one boundary component.

Equations defining the jumping loci for the first cohomology group of one-dimensional representations of a finitely presented group $\Gamma $ can be effectively computed using Fox calculus. In this paper, we give an exposition of Fox calculus in the language of group cohomology and in the language of finite abelian coverings of CW complexes. Work of Arapura and Simpson imply that if $\Gamma $ is the fundamental group of a compact Kähler manifold, then the strata are finite unions of translated affine subtori. It...

M. Bestvina has shown that for any given torsion-free CAT(0) group G, all of its boundaries are shape equivalent. He then posed the question of whether they satisfy the stronger condition of being cell-like equivalent. In this article we prove that the answer is "Yes" in the situation where the group in question splits as a direct product with infinite factors. We accomplish this by proving an interesting theorem in shape theory.

We show that the amalgamated free products of two free groups over a cyclic subgroup admit amenable, faithful and transitive actions on infinite countable sets. This work generalizes the results on such actions for doubles of free group on two generators.

We study under which condition an amalgamated free product or an HNN-extension over a finite subgroup admits an amenable, transitive and faithful action on an infinite countable set. We show that such an action exists if the initial groups admit an amenable and almost free action with infinite orbits (e.g. virtually free groups or infinite amenable groups). Our result relies on the Baire category Theorem. We extend the result to groups acting on trees.

We extend Gromov's notion of asymptotic dimension of finitely generated groups to all discrete groups. In particular, we extend the Hurewicz type theorem proven in [B-D2] to general groups. Then we use this extension to prove a formula for the asymptotic dimension of finitely generated solvable groups in terms of their Hirsch length.

We prove that the boundary of a right-angled hyperbolic building is a universal Menger space. As a consequence, the 3-dimensional universal Menger space is the boundary of some Gromov-hyperbolic group.